The empirical rule (or 68-95-99.7 rule) provides an approximation of how data is distributed in a normal (bell-shaped) distribution. According to the empirical rule:

• 68% of the data falls within 1 standard deviation of the mean.
• 95% of the data falls within 2 standard deviations of the mean.
• 99.7% of the data falls within 3 standard deviations of the mean.

Given information:

• Mean body temperature $\mu = 98.03^\circ F$
• Standard deviation $\sigma = 0.42^\circ F$

Part a.

For body temperatures within 1 standard deviation of the mean, the range is: $\mu - \sigma = 98.03 - 0.42 = 97.61^\circ F$ $\mu + \sigma = 98.03 + 0.42 = 98.45^\circ F$ Thus, the temperatures range from $97.61^\circ F$ to $98.45^\circ F$.

According to the empirical rule, approximately 68% of healthy adults will have body temperatures within 1 standard deviation of the mean.

Part b.

For body temperatures between $96.77^\circ F$ and $99.29^\circ F$, we need to check how many standard deviations these values are from the mean.

1. Calculate the number of standard deviations for $96.77^\circ F$: $\frac{96.77 - 98.03}{0.42} = \frac{-1.26}{0.42} = -3 \text{ standard deviations}$

2. Calculate the number of standard deviations for $99.29^\circ F$: $\frac{99.29 - 98.03}{0.42} = \frac{1.26}{0.42} = 3 \text{ standard deviations}$

According to the empirical rule, approximately 99.7% of the data falls within 3 standard deviations of the mean.

Summary:

• a. Approximately 68% of healthy adults have body temperatures between 97.61°F and 98.45°F.
• b. Approximately 99.7% of healthy adults have body temperatures between 96.77°F and 99.29°F.

Do you want more details or have any questions?

Related questions:

1. What is the approximate percentage of adults with temperatures between 98.45°F and 99.29°F?
2. How does the empirical rule apply to distributions with different shapes?
3. What percentage of healthy adults will have a temperature above 99.29°F?
4. How do outliers affect the empirical rule?
5. How does this distribution compare to the standard normal distribution?

Tip: When using the empirical rule, always check that the data is approximately normal for the rule to apply effectively.